
Signature \((\cdot ,D,\cap )\) . The resulting variety of enriched left restriction semigroups was first axiomatised in [1]} and [2]}.

Signature \((\cdot ,D,-)\) . These have not previously been considered. Note that the class of minus-semigroups is a proper quasivariety, as noted earlier. When we introduce left restriction semigroup structure to induce the right normal band structure, then adding an identity element 1 with respect to the composition operation, we obtain the class of modal restriction semigroups in which \(A(x)=1-x\) , as in [3]}. These form a proper quasivariety, as shown there, so the current class must also.

Signature \((\cdot ,D,-,\cap )\) . Not previously considered, but a variety, following Proposition REF . With an identity element under composition added, we obtain modal restriction semigroups with intersection as in [3]}.

Signature \((\cdot ,D,-,\sqcup )\) . Again, not previously considered, but yields a finitely based variety, as did \((\cdot ,\circ ,-,\sqcup )\) . With an identity element under composition added, we obtain modal restriction semigroups with preferential union as in [3]}.

Signature \((\cdot ,D,-,\diamond )\) . This is a new signature not previously considered, even with an identity element present, and is a finitely based variety due to Proposition REF . With a composition identity added, the resulting class of modal restriction semigroups is a new one enriched with the addition of update.

Signature \((\cdot ,D,\cap ,\sqcup )\) . This is a signature not previously considered, again a finitely based variety. Since minus is not present, adding a composition identity does not give a class of enriched modal restriction semigroups.

Signature \((\cdot ,D,\cap ,\diamond )\) . Again, a new signature. We are uncertain whether it is a variety.

